Solution Of Variational Inequality Research Articles

Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearityht=∇⋅(1|∇h|∇eδEδh)=∇⋅(1|∇h|∇e−∇⋅(∇h|∇h|)) where total energy E=∫|∇h| is the total variation of h. Using a logarithmic correction for total energy E=∫|∇h|ln⁡|∇h|dx and gradient flow structure with a suitable defined functional, we prove the one dimensional evolution variational inequality solution preserves a positive gradient hx which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity hxx+ to happen.

Bregman Extragradient Method with Monotone Rule of Step Adjustment*

A new extragradient-type method is proposed for approximate solution of variational inequalities with pseudo-monotone and Lipschitz-continuous operators acting in a finite-dimensional linear normed space. The method uses Bregman divergence (distance) instead of Euclidean distance and a new adjustment of step size, which does not require knowledge of the Lipschitz constant of the operator. In contrast to the previously used rules for choosing the step size, the method proposed in the paper does not perform additional calculations for the operator values and prox-map. A theorem on the convergence of the method is proved.

Convergence of Two-Stage Method with Bregman Divergence for Solving Variational Inequalities*

A new two-stage method is proposed for the approximate solution of variational inequalities with pseudo-monotone and Lipschitz-continuous operators acting in a finite-dimensional linear normed space. This method is a modification of several well-known two-stage algorithms using the Bregman divergence instead of the Euclidean distance. Like other schemes using Bregman divergence, the proposed method can sometimes efficiently take into account the structure of the feasible set of the problem. A theorem on the convergence of the method is proved and, in the case of a monotone operator and convex compact feasible set, non-asymptotic estimates of the efficiency of the method are obtained.

Nash and Wardrop Equilibria in Aggregative Games With Coupling Constraints

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibrium. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.

On the existence of solutions of variational inequalities in nonreflexive Banach spaces

We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.

Gradient flow approach to an exponential thin film equation: global existence and latent singularity

In this work, we study a fourth order exponential equation,ut= Δe−Δuderived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.

Viscosity Methods for Approximating Solutions of Variational Inequalities for Asymptotically Nonexpansive Mappings

Viscosity Methods for Approximating Solutions of Variational Inequalities for Asymptotically Nonexpansive Mappings

The Valuation of American Passport Options: A Viscosity Solution Approach

The passport option, introduced and marketed by Bankers Trust, is a call option on the balance of a trading account. This paper concerns the American passport option. We rigorously establish the mathematical foundation for pricing the American passport option. We derive the pricing equation, using the dynamic programming principle, and prove that the option value is a viscosity solution of variational inequality, which is a fully nonlinear equation. We also establish the comparison principle, which yields uniqueness of the viscosity solution. Moreover, we prove convexity-preserving property for the viscosity solution. In addition, we obtain further properties of the optimal exercise boundary. Finally, we give several numerical examples and financial analysis.

Homogenization of some degenerate pseudoparabolic variational inequalities

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.

Stochastic approximation results for variational inequality problem using random-type iterative schemes

ABSTRACTReal world problems are embedded with uncertainties. Therefore, to tackle these problems, one must consider probabilistic nature of the problems both in modeling and solution. In this work, concepts of convergence of the solution of variational inequality in classical functional analysis are extended to a stochastic domain for a random Mann-type iterative and Ishikawa-type iterative schemes in a Banach space. A mean square convergence result is proved for this extension.

A multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the [formula omitted]-Laplacian

In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the p-Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems.

English

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.

A note on strongly nonlinear parabolic variational inequalities

Abstract We prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

A New Simple Parallel Iteration Method for a Class of Variational Inequalities

In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of m-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm.

CQ method for approximating fixed points of nonexpansive semigroups and strictly pseudo-contractive mappings

We use the CQ method for approximating a common fixed point of a left amenable semigroup of nonexpansive mappings, an infinite family of strictly pseudo-contraction mappings and the set of solutions of variational inequalities for monotone, Lipschitz-continuous mappings in a real Hilbert space. Our results are a generalization of a result announced by Nadezhkina and Takahashi [N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241] and some other recent results.

Approximation of a simply supported plate with obstacle

We discuss an algorithm for the solution of variational inequalities associated with simply supported plates in contact with a rigid obstacle. Our approach has a fixed domain character, uses just linear equations and approximates both the solution and the corresponding coincidence set. Numerical examples are also provided.

Approximating fixed point solutions of variational inequalities using explicit iterations for asymptotically nonexpansive semigroup of mappings in Banach spaces

Approximating fixed point solutions of variational inequalities using explicit iterations for asymptotically nonexpansive semigroup of mappings in Banach spaces

Discrete-time projection neural network methods for computing the solution of variational inequalities

Discrete-time projection neural network methods for computing the solution of variational inequalities

Strong convergence results for variational inequalities and fixed point problems using modified viscosity implicit rules

Our aim in this paper is to introduce a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverse-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. Some strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As an application, we give an algorithm to solve fixed point problems for nonexpansive mappings, variational inequality problems and equilibrium problems in Hilbert spaces. Finally, we give one numerical example to illustrate our convergence analysis.

The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set

ABSTRACTWe introduce a new iteration algorithm for solving the Ky Fan inequality over the fixed point set of a nonexpansive mapping, where the cost bifunction is monotone without Lipschitz-type continuity. The algorithm is based on the idea of the ergodic iteration method for solving multi-valued variational inequality which is proposed by Bruck [On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), pp. 159–164] and the auxiliary problem principle for equilibrium problems P.N. Anh, T.N. Hai, and P.M. Tuan. [On ergodic algorithms for equilibrium problems, J. Glob. Optim. 64 (2016), pp. 179–195]. By choosing suitable regularization parameters, we also present the convergence analysis in detail for the algorithm and give some illustrative examples.